Problem: What is the greatest common factor of $3x^4$, $15x^3$, and $21x^2$ ?
Answer: Let's factor each monomial to its prime factors: $\begin{aligned} 3x^4&=(3)(x)(x)(x)(x) \\\\ 15x^3&=(3)(5)(x)(x)(x) \\\\ 21x^2&=(3)(7)(x)(x) \end{aligned}$ We want the largest set of factors that's included in all three monomials. All of the monomials have one factor of $ 3$ and two factors of $ x$ : $\begin{aligned} 3x^4&=( 3)( x)( x)(x)(x) \\\\ 15x^3&=( 3)(5)( x)( x)(x) \\\\ 21x^2&=( 3)(7)( x)( x) \end{aligned}$ This is the greatest common factor: $( 3)( x)( x)=3x^2$